On invariant measures of finite affine type tilings

Details On invariant measures of finite affine type tilings On invariant measures of finite affine type tilings

2004-12-15

arxiv.org/abs/math/0412290v1

Mathematics

Dynamical Systems

18 pages, 3 figures, submit

Scientific paper

In this paper, we consider tilings of the hyperbolic 2-space, built with a finite number of polygonal tiles, up to affine transformation. To such a tiling T, we associate a space of tilings: the continuous hull Omega(T) on which the affine group acts. This space Omega(T) inherits a solenoid structure whose leaves correspond to the orbits of the affine group. First we prove the finite harmonic measures of this laminated space correspond to finite invariant measures for the affine group action. Then we give a complete combinatorial description of these finite invariant measures. Finally we give examples with an arbitrary number of ergodic invariant probability measures.

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